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Math Help - Question related to Weierstrass criteria for uniformly convergent series of functions

  1. #1
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    Question Question related to Weierstrass criteria for uniformly convergent series of functions

    hello to all

    I have some issue of understanding this criteria... it says:

    Let

    \displaystyle (a_n)_{n\in \mathbb{N}

    be positive numerical sequence for which is worth that for almost every

    \displaystyle  n \in \mathbb{N}

    and every

    \displaystyle  x\in A

    is satisfied that

    \displaystyle  |f_n (x)|\le a_n

    then if series:

    \displaystyle \sum_{n=1} ^{+\infty} a_n

    is convergent then series (of functions) :

    \displaystyle \sum_{n=1} ^{+\infty} f_n(x)

    is uniformly convergent on set A.
    my question is why it is "uniformly" convergent ? I just can't see the fact why we have here uniformly convergent series...

    Thanks for any help
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  2. #2
    Member
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    The Wikipedia article on it explains it nicely: Uniform convergence - Wikipedia, the free encyclopedia
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  3. #3
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    I don't have a problem with understanding the meaning of uniform convergence... I have problem with this criteria... perhaps i was not clear about it very good... I'm also having problem with English hehehe

    It's like this.... Why with assumption that (a_n) is sequence of positive numbers... and if ... (everything in the criteria) .... how someone can conclude that series of functions uniformly converge ... because of what ? Is it because of absolutely convergence of numerical series... ? or what ? how do I know that It's uniformly converge and not just by points...
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  4. #4
    Senior Member yeKciM's Avatar
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